Likelihood Ratio: Formula and How to Calculate It

A likelihood ratio compares how well two competing hypotheses explain observed evidence: divide the probability of the evidence under one hypothesis by its probability under the other. Enter those two probabilities to calculate the ratio, then use its direction and size to see which hypothesis the evidence supports more strongly.

Formula

The likelihood ratio for evidence E comparing hypotheses H1 and H2 is:

LR = P(E | H1) / P(E | H2)

• P(E | H1) is the probability of observing the evidence if H1 is true.
• P(E | H2) is the probability of observing the evidence if H2 is true.
• LR is the likelihood ratio for H1 relative to H2. Probabilities are dimensionless numbers from 0 to 1, so the ratio has no unit.

The calculator uses p1 for P(E | H1) and p2 for P(E | H2). It also shows the reciprocal, which compares H2 with H1. A likelihood is not the same as the probability that a hypothesis is true: it describes how probable the evidence is under a hypothesis.

Worked Example

Suppose the evidence has probability 0.8 under H1 and 0.2 under H2. These are the calculator’s default values.

1. Substitute the values: LR = 0.8 / 0.2.
2. Divide: LR = 4.00.
3. For the reverse comparison: 0.2 / 0.8 = 0.25.

An LR of 4.00 means the evidence is four times as probable under H1 as under H2. The reciprocal, 0.25, expresses the same comparison in the other direction: the evidence is one quarter as probable under H2 as under H1.

How to Interpret a Likelihood Ratio

Interpret the ratio only after checking which hypothesis is in the numerator. If LR is greater than 1, the evidence is more probable under H1; if it is less than 1, it is more probable under H2. An LR of exactly 1 means the evidence has equal probability under both hypotheses and therefore does not distinguish between them.

• LR = 5: the evidence is five times as probable under H1 as under H2.
• LR = 0.2: the evidence is five times as probable under H2 as under H1.
• LR = 1: the evidence is equally probable under both hypotheses.

There is no universal cutoff at which a likelihood ratio proves a hypothesis. Whether a value is important depends on the setting, the quality of the probability estimates, and the other evidence. A large ratio can strongly favor one hypothesis over another without establishing that the favored hypothesis is true.

Likelihood Ratios and Prior Odds

A likelihood ratio compares evidence, not starting beliefs. To update the relative odds of two hypotheses, multiply their prior odds by the likelihood ratio:

Posterior odds for H1 over H2 = prior odds for H1 over H2 × LR

For example, if the prior odds favor H2 by 2 to 1, the odds for H1 over H2 are 1/2. With an LR of 4, the posterior odds become (1/2) × 4 = 2, or 2 to 1 in favor of H1. This calculation assumes the hypotheses are being compared consistently and that the likelihoods correctly describe the evidence. Converting posterior odds into a probability also requires a clearly defined set of alternatives.

For several pieces of evidence, likelihood ratios can be multiplied when the evidence items are conditionally independent under each hypothesis. If they are dependent, multiplying separate ratios as though they were independent can overstate the evidence. The joint probabilities, or a model that accounts for dependence, are needed instead.

Common Problems and Limits

The arithmetic is simple; choosing defensible probabilities is usually the harder part. Probabilities should refer to the same precisely stated evidence and be calculated under the same competing hypotheses. Vague hypotheses or changing the definition of the evidence can make a numerical comparison misleading.

• Reversing the comparison: swapping numerator and denominator gives the reciprocal. State clearly whether the ratio is H1 over H2 or H2 over H1.
• Confusing evidence probability with hypothesis probability: P(E | H) is not P(H | E). The latter depends on prior odds as well as the likelihood of the evidence.
• Using rounded or incompatible inputs: round only after division when possible. Ensure both inputs describe the same event and use comparable assumptions.
• Dividing by zero: if P(E | H2) is zero, the ratio is undefined or treated as infinite in an extended comparison. A zero may reflect a model that rules out the evidence, or an estimate with insufficient data; it does not automatically prove H1.
• Treating a large ratio as certainty: a ratio measures relative support from specified evidence. It does not account for every alternative explanation or fix flawed assumptions.

A likelihood ratio is a comparison, so it is not an absolute measure of how common evidence is. For instance, probabilities of 0.8 and 0.2 produce the same ratio as probabilities of 0.08 and 0.02, even though the evidence itself is less probable in both cases in the second comparison. The ratio alone does not convey that distinction.

Frequently Asked Questions

What does a likelihood ratio of 1 mean?
It means the observed evidence is equally probable under the two hypotheses being compared. The evidence does not favor one over the other through this ratio.

Can a likelihood ratio be less than 1?
Yes. A value below 1 means the evidence is less probable under the numerator hypothesis than under the denominator hypothesis. Taking the reciprocal gives the comparison in the opposite direction.

Is a likelihood ratio the same as a probability?
No. A probability is between 0 and 1, while a likelihood ratio can be any nonnegative value and may exceed 1. It compares two probabilities rather than giving the probability that a hypothesis is true.

How do I calculate a likelihood ratio from counts?
Estimate the probability of the same evidence under each hypothesis using the relevant counts or data, then divide the first probability by the second. The data must support both estimates, and the method should account for sampling uncertainty where relevant.

How do I combine likelihood ratios?
Multiply ratios for separate evidence only when the evidence is conditionally independent under both hypotheses. Otherwise, calculate a ratio from the joint evidence or use a model that represents the dependence.

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in p1 | Evidence probability under hypothesis 1 | - | 0.8 | 0
in p2 | Evidence probability under hypothesis 2 | - | 0.2 | 0
out lr | Likelihood ratio, H1 over H2 | - | p1 / p2 | 2
out reciprocal | Likelihood ratio, H2 over H1 | - | p2 / p1 | 2